The determination of thermal residual stresses in unidirectional and cross-ply titanium matrix composites using an etch removal method

Gerald CR Watt1,2, Andrew D Crocombe1, Stephen L Ogin1, and Stephen Kyle-Henney2

1Faculty of Engineering and Physical Sciences, University of Surrey, Guildford, Surrey, UK2Composites Group, TISICS Ltd, Farnborough, Hampshire, UK

AbstractRecent work has shown that a simple rule-of-mixtures approach may be used to predict the stress strain behaviour of a cross-ply metal matrix composite (MMC) laminate. However, the low-strain behaviour was not predicted accurately, probably because thermal residual stresses are obviously not included in such an approach. To increase the understanding of the limitations of the rule-of-mixtures approach for predicting the stress-strain response, the residual strain-state of the fibre reinforcement has been determined using an etching technique (henceforth referred to as the ‘total etch removal method’), and results have been compared both with finite element modelling and with thermal residual strain measurements derived from stress-strain curves. The results show that the residual strain distribution in a cross-ply composite may be more complex than previously thought, with the fibres in internal 00 plies having considerably higher thermal residual strains than fibres in external plies. The results confirm that the rule-of-mixtures approximation can be used, with some reservations with regard to the low strain behaviour.

KeywordsMetal matrix composites (MMCs), Residual stress, Finite element analysis (FEA), Micro-mechanics.

IntroductionRecent interest from the space industry has meant that continuous fibre metal matrix composites (MMCs) are now being considered for a wider variety of components than simply loaded booms and struts. In particular, the merits of silicon carbide (SiC) continuous fibre reinforced pressure vessels are currently being assessed [1, 2]. At present, a wide range of monolithic tank constructions are in service within the commercial satellite industry but, compatible alloys required for the long-term storage of

propellants do not have the desired specific strength and stiffness, and MMCs offer a potential advantage over conventional materials.

To contain loading from internal pressurisation, MMC layups must be able to withstand considerable biaxial stresses. However, it is well established that the transverse properties of unidirectional MMCs are poor in comparison to their parent alloys [3] as a result of the weak interface that exists between the fibre and matrix [4-6]. Consequently, where bi-axiality is required, a layup of mixed directionality is necessary.

To date, a limited amount of research has been conducted on continuous fibre bi-axial layups, and the role and contribution of thermal residual stresses to the mechanical performance is generally only theoretical in nature [7, 9-10]. Ultimately, the design of structures for pressure containment requires the behaviour of biaxial and multiaxial MMC layups to be better understood and more experimental work is required to substantiate FEA results. Previous work conducted on the tensile response of cross-ply TMCs by Watt et al [12] has shown that a good prediction of cross-ply stress-strain response up to failure could be obtained by averaging the stress strain data from the component 0° and 90° laminates (i.e. a “rule-of-mixtures” approach). Figure 1 shows typical stress-strain curves for a 00 ply, a 900 ply and a cross-ply laminate ([0/90]2s), together with the rule-of-mixtures prediction for the cross-ply laminate.

Figure 1. Prediction of the stress strain behaviour of a [0/90]2s TMC in tension from the stress-strain behaviour of UD plies for longitudinal and transverse loading, using the rule-of-mixtures.

An inaccuracy associated with this prediction was the inability to account for the premature onset of non-linearity in the [0/90]2s coupons, which was suggested to be a consequence of the thermal residual stress state of the matrix in the laminate. In order to investigate and quantify the residual stresses, a version of the etch removal method (called here the “total etch removal method”) has been employed for the determination of residual stresses in MMCs. Previous researchers have experimented with variations of the etch removal method to release the induced compressive thermal strain in the fibres [13-19]. For example, an etching agent was dropped onto a site perpendicular to the fibre ends and where the matrix was removed, fibre extension was measured (see Fig 2) [13-15].

Figure 2. Etch removal technique from Pickard and Miracle [14].

A modification of this technique was to use a ‘selective etch’ over a bundle of 20 to 30 fibres contained in a narrow slit of MMC by masking off either side of the strip before exposure to the etchant [16-18]. The resulting fibre height change (illustrated in Fig 3) could then be measured via travelling, or confocal, microscopy and axial strain calculated based on measured extensions. In more recent years, a method of back calculating hoop stresses with the Lamé equations for an axisymmetric cylinder model [19] has been tested on existing selective matrix etch test data [16,17] and validated with FEA.

Figure 3. Etch removal technique with wire eroded slit and masked off areas from Ramamurty et al [16].

One of two finite element approaches is generally applied to the modelling of MMCs, either micro-modelling or macro-modelling [20]. The most-widely used method for the detailed modelling of residual stresses within a representative composite model is micro-modelling, where a small section of the overall body is created and symmetry and boundary conditions are used. Axisymmetric or concentric cylinder assemblage models (CCMs) provide good approximations for unidirectional composites, but to model more complex layups (such as the cross-ply laminates used in this work) a 3D unit cell/representative volume element (RVE) is required. Uniaxial stress models (USMs) and concentric cylinder models were investigated by Neu et al [7], who incorporated viscoplastic matrix effects; they found the maximum difference between axial residual stress in the models to be 15% at room temperature. However, it was pointed out by Zhang et al [8] that the axial stress component is not a direct function of fibre spacing, so CCMs and USMs should produce similar results.

The primary purpose of the current paper is to use a new variation of the etch removal method to determine the residual thermal stresses in the axial fibres of a cross-ply titanium matrix composite and to validate the use of the rule-of-mixtures approximation as a summation of the behaviour of 0° and 90° plies tested separately. Measurements from unidirectional composites are used to predict the lock-in temperature, enabling comparisons with mechanical test results and finite element modelling predictions.

Experimental methods

Coupon fabrication

Fabrication of the unidirectional [0]8 and cross-ply [0/90]2s TMCs, and monolithic coupons, for testing followed the standard TISICS procedure: Titanium foil (Ti-3Al-2.5V) was cut to size on a guillotine and degreased with acetone. Sheets of aligned fibre containing a fugitive binder were then sandwiched between the degreased foils until the desired ply thickness was reached (eight plies in this case). The fibre was produced in-house (via a chemical vapour deposition process) before being filament wound and sprayed with a polymeric binder to maintain uniform fibre spacing; for all the coupons in this work a 140 μm diameter SiC monofilament was used (SM3156), and the titanium foil thickness was also 140 μm thick. The foil/fibre layups were contained between yttria release agent coated steel packing plates and placed into a stainless steel can. The layups fabricated were [0]8 for the unidirectional panels and [0/90]2s for the cross-ply panels. The can was welded to contain the layup and then evacuated through a steel degas pipe. A cycle of heating was used to remove the fugitive binder while evacuating the can at moderate to high vacuum levels. Subsequently, the degas pipe was sealed, containing the high internal vacuum (approximately 10-3 Pa). The can was then subject to a hot isostatic press (HIP) cycle with a maximum temperature of around 950 ˚C and 100 MPa pressure. After processing, the stainless steel can was mechanically removed and the yttria coated packing plates were released from the consolidated panel. To produce monolithic coupons of the titanium matrix alloy, ten sheets of 140 μm thick cold-rolled Ti-3Al-2.5V

foil were diffusion bonded into a flat panel using the same manufacturing methodology, but without the addition of fibre.

Coupons conforming to ASTM D3552-12 were cut from their parent panels using wire electro-discharge machining (EDM) and the edges were wet ground with a diamond impregnated grinding wheel to remove the recast layer produced by the EDM. Monolithic aluminium end tabs were then affixed to the tensile coupons with a structural adhesive (3M 9323 B/A) to prevent damage from the knurled wedge grips when testing. The compressive coupons were supplied with no tabs to fit the IITRI test fixture (as shown in ASTM D3410). Consolidated thicknesses for the unidirectional and cross-ply coupons were 1.897 mm ± 5 μm and 1.948 mm ± 7 μm, respectively.

Mechanical testing of coupons

An electro-mechanical Instron 4505 test frame (100 kN load cell) was used to load parallel sided coupons 20 mm wide x 170 mm long (conforming to ASTM D3552 guidelines) to failure in tension at a cross-head displacement rate of 2 mm min-1 and longitudinal strain gauges (TML BFLA-2-5) were used to measure the strain. Ambient temperature mechanical test data was obtained for both unidirectional [0]8 and cross-ply [0/90]2s TMCs. For the compression tests, parallel sided coupons measuring 20 mm wide x 150 mm long were tested in an Illinois Institute of Technology Research Institute (IITRI) style fixture by Airbus Defence and Space. The IITRI rig supported the coupon from out of plane bending thus preventing premature buckling. The specimens were gripped over 70 mm at each end, leaving a gauge length of 10 mm between the two clamping wedges, strain gauges on both sides of the coupon gauge section were used. The tensile and compressive coupon tests had the same nominal fibre volume fraction of 35% as the TMC specimens used in the etch removal technique.

Total etch removal method (TERM)

Two sets of TMC coupons of nominal dimensions 20 mm wide x 100 mm long were removed from their parent panels using EDM. The first specimens analysed were unidirectional [08] TMC with TISICS SM3156 fibre in a Ti-3Al-2.5V matrix. The second set of specimens were cross-ply [0/90]2s TMC with the same nominal volume fraction and fibre/matrix combination. A surface grinder with a diamond grinding wheel was used to wet grind the edges of the coupons, thereby removing the recast layer left from EDM. Once all coupons had been ground parallel, dimensional measurements were taken on a Nikon V12 profile projector (shadowgraph) to determine the long axis fibre length. Volume fraction measurements of the composites were made using optical microscopy images of polished mounted specimens (Figures 4 & 5) taken from the parent panels and the average volume fibre fraction was found to be 35%.

Figure 4. A micrograph of a mounted and polished uniaxial TMC specimen viewed on a plane perpendicular to the fibre axes.

Figure 5. A micrograph of a mounted and polished cross-ply TMC specimen, viewed on a plane perpendicular to the 00 fibres (note: blemishing is due to the etchant).

Prior to etching the specimens, measurements were performed with the shadowgraph in reflected light mode with a 100x objective lens to measure the edge-to-edge distance along the coupon parallel to the 00 fibres. Connected to the traversing table was a Metronic 200 digital read-out which output X and Y coordinates to a precision of 1 µm. The shadowgraph was used again once the fibres had been liberated from the matrix with a strong etch consisting of a 5% HF / 23% HNO3 solution. The solution created for etching was not, however, of sufficient strength to cause damage to the fibres but acted to dissolve the matrix completely. Samples of approximately one hundred full-length fibres from each coupon were randomly selected, and these were placed individually into a milled alignment block and the fibre length was measured. The purpose of the block was to provide an index point and to support and align the fibres during measurement under the shadowgraph, thus removing inaccuracies resulting from possible fibre curvature.

Mechanical test and TERM results

Derivation of residual stresses from the unidirectional composite mechanical test data

Results from the unidirectional TMC coupon testing are shown in Figure 6, where the solid lines represent tensile behaviour and the dashed lines denote compressive stress-strain behaviour. A clear change of gradient or ‘knee’ is evident in both sets of curves and this represents the point at which macroscopic yield of the matrix occurs. Once the matrix shows plastic behaviour, the instantaneous modulus of the composite is dominated by the presence of the fibres, which exhibit linear elastic behaviour up to failure.

Figure 6. Compressive and tensile behaviour of a unidirectional SM3156 TMC.

The difference in measured strain between the tensile and compressive knees in the stress-strain curves is known to be a consequence of the residual stress state imparted upon cool-down from processing [21, 22]; the matrix reaches a stress state sufficient to cause macroscopic yielding and a noticeable change in instantaneous modulus occurs. It should be noted that the required stress to yield the ‘as-HIPed’ condition monolithic matrix alloy (Ti-3Al-2.5V) remains virtually identical in both tension and compression loading at about 0.5% strain (as shown by Figure 7). However, premature failure of the strain gauges used for strain measurement meant that strain-to-failure of the coupons was not recorded.

Figure 7. Compressive and tensile behaviour of ‘as-HIPed’ monolithic Ti-3Al-2.5V coupons. Dashed lines show the compressive specimen, solid lines are the tensile specimen.

Due to the CTE mismatch between the fibres and the matrix in the unidirectional MMC, the fibres are under a residual state of compression and the matrix is in a state of residual tension. When compression testing of the composite begins, the state of residual tension in the matrix must be fully reversed before an axial compressive stress can be induced in the matrix, and yield reached. Consequently, matrix yield and change of instantaneous modulus in the unidirectional TMC, occur at a higher applied strain than for that of the monolithic matrix alloy (Fig 7) in compression (εcmy = - 0.75%). Similarly, the applied tensile strain required to cause yield in a TMC is lower than that of the monolithic alloy as the matrix is already in a state of residual tension (εcmy= 0.35%).

The mechanical test results enable a simple derivation of the residual strain in the fibres for the unloaded unidirectional composite to be made. Assuming that matrix yield is a consequence of axial stress, rather than von Mises stress, the strain that would need to be applied to the unidirectional (UD) composite to achieve zero matrix stress is¿ε tmy∨−¿ εcmy∨¿2. Here, εcmy refers to the strain required for composite compressive matrix yield, and εtmy denotes the strain for yield of the matrix in tension. At zero matrix stress, all the load is carried by the fibres, giving the fibre stress, which is compressive, as

σ f =[|εtmy|−|εcmy|2 ] Ec

V f (1)

where Vf is the volume fraction of fibres and Ec is the composite modulus. Unloading the

composite from the compressive strain of |εtmy|−|ε cmy|2

to zero strain, reduces the

compressive strain in the fibre such that for the unloaded composite, the fibre residual stress, σresf, is

σ resf=[|ε tmy|−|εcmy|2 ][ Ec

V f−E f ] (2)

where Ef is the fibre modulus. The corresponding residual stress in the matrix, which is tensile, is given by

σ resm=[|σcmy|−|σ tmy|2 ][ Em

Ec ] (3)where, σcmy is the stress to macroscopic matrix yield in compression loading (onset of non-linearity) and σtmy is the onset of non-linearity in the tensile regime; Em is the matrix Young’s modulus. It should be noted that equation (3) is identical to the expression derived by Newaz and Majumdar [23]. From equation (2), the fibre residual strain, εresf, for the unloaded UD MMC, is then

ε resf=[|εtmy|−|εcmy|2 ][ Ec

V f Ef−1] (4)

Tension and compression test data for the fibre volume fraction of Vf = 0.35 produced a composite modulus, Ec = 213 ± 5 GPa. A value for the Young’s modulus of the SM3156 monofilament is more problematic. A similar fibre (SM2156) has a modulus of about 385 GPa [24] but the SM2156 has more free carbon in the SiC region and would be expected to exhibit a lower modulus than SM3156 which maintains a stoichiometric balance. Applying a rule-of-mixtures approach to the UD tensile data obtained in this work suggests a SM3156 modulus of about Ef = 400 GPa, which does not seem unreasonable. Using these values, equation (4) provides a residual axial thermal strain in the fibres, as a consequence of thermal stresses, of εresf = - 0.104%. This value will be compared with the results of the etch removal experiments in the next section. The calculated strain would result in a residual axial fibre stress of 416 MPa and a matrix stress of 224 MPa to maintain equilibrium.

Derivation of residual stresses from the cross-ply composite mechanical and physical property data

Figure 8. Compressive and tensile behaviour of a cross-ply SM3156 TMC coupons.

Typical results from the cross-ply TMC coupon testing are shown in Figure 8, where the solid lines represent tensile behaviour and the dashed lines denote compressive stress-strain behaviour. A change of gradient or ‘knee’ is evident in both sets of curves; however, unlike the unidirectional stress strain curves in which the knee is primarily a result of plastic deformation in the matrix [25], the tensile knees in the cross-ply TMCs are partly the consequence of matrix plasticity but also of additional mechanisms (such as fibre/matrix debonding). The fibre debonding in the transverse plies causes a reduction in the instantaneous modulus in tension from a very small strain. Replication work conducted by Johnson et al [26] has confirmed that debonding occurs in the transverse plies of similar [0/90]2s cross-ply TMCs at stresses above the initial knee (0.08%) in the tensile stress strain response. Other researchers, such as Xia et al [27], have used micromodelling techniques to investigate the propagation of debonding in cross-ply TMCs suggesting that an induced strain of approximately 0.06% is sufficient to initiate debonding of the transverse plies. Consequently, the knees in the stress-strain curves

cannot here be employed to determine the thermal residual strains and a different approach must be used to predict the residual fibre strain in the cross-ply composites.

Using mechanical property data from the unidirectional and transverse laminates, along with known thermal expansion coefficients, the residual fibre strain can be determined, again using a simple one-dimensional model, by considering the force-balance parallel to the 00 fibres. Firstly, the strain in the laminate (εrx,0) parallel to the 00 fibres is calculated from the force balance for the axial and transverse plies, taking into account strain compatibility i.e.

ε rx , 0=E90 ∆T (α90−α0)

E0+E90 (5)

Where E0 is the axial ply modulus, E90 is the transverse ply modulus, ΔT refers to the change in temperature, α0 defines the axial ply CTE and α90 defines the transverse CTE. The residual tensile fibre strains, εresf, for the 00 fibres within the axial plies of the cross-ply laminate is then found to be:

ε resf=E0 εrx , 0+ Em(1−V f )∆ T (αm−α f )

Ef V f +Em(1−V f )

(6)Here, Em is the monolithic Ti-3Al-2.5V modulus, Ef denotes the fibre modulus, αf and αm are the fibre and matrix CTEs, respectively. Equation (6) predicts a residual 00 fibre strain in the cross-ply laminate of εresf = - 0.17%. It should be noted that this approach assumes that the strains are the same for the innermost and outermost axial (00) plies.

Derivation of residual stresses from the TERM of unidirectional composites

Figure 9. Gaussian distribution of released fibre strains from the unidirectional coupons.

Experimental results for the residual strains measured from three UD coupons (using the TERM method) are shown in Figure 9, with approximately 100 fibres analysed individually in each case; the data points for each set of results has been joined together for clarity. The strain was calculated from a comparison of the free fibre lengths with the coupon lengths. Averaged over the three coupons, the fibre residual strain was found to be εresf = - 0.13% ± 0.02 (see Table 1). This is 25% higher than the strain found from the analysis of the mechanical test data in Section 3.1 (i.e. - 0.104%). A fibre residual strain of - 0.13% corresponds to an average [compressive] residual axial stress of about 520 MPa in the fibres, and 280 MPa in the matrix.

Table 1. Results of the total etch removal method on unidirectional TMC.

UD Coupon 1 UD Coupon 2 UD Coupon 3

Average fibre extension (μm) 137 ± 17 125 ± 13 129 ± 23

Longitudinal fibre strain (%) 0.14 ± 0.02 0.13 ± 0.01 0.13 ± 0.02

Longitudinal fibre stress (MPa) 546 ± 68 511 ± 52 526 ± 96

Longitudinal matrix stress (MPa) 294 275 283

Derivation of residual stresses from the TERM method applied to cross-ply TMC coupons

Figure 10. Gaussian distribution of released fibre strain from the cross-ply coupons.

The total etch removal method was also applied to cross-ply coupons. Again, samples of one hundred axial fibres were selected from each of the three coupons for measurement under the shadowgraph. As the residual stress state in the cross-ply will be different from the UD specimen, a different fibre residual strain was expected. Figure 10 shows the residual strains for the three specimens and it is clear that there is a bi-modal distribution. Average results for the three coupons are shown in Table 2 where the average residual strain is about - 0.32 ± 0.08%. However, Figure 10 shows there are peak average strains

at strains of about - 0.23 ± 0.03% and - 0.37 ± 0.04%. The presence of the two peaks warranted further investigation.

Table 2. Results of the total etch removal method on cross-ply TMC coupons.

CP Coupon 1 CP Coupon 2 CP Coupon 3

Average fibre extension (μm) 319 ± 100 339 ± 43 355 ± 72

Longitudinal fibre strain (%) 0.320 ± 0.100 0.340 ± 0.044 0.295 ± 0.072

Longitudinal fibre stress (MPa) 1280 ± 400 1360 ± 176 1180 ± 288

A parallel-sided coupon of cross-ply composite (approximately 50 mm long by 20 mm wide), from the same parent panel as the previous coupons, was prepared. Two parallel slits were cut with a length of 24 mm into the panel, equidistant from the centre, with a Struers Accutom and a 0.6 mm thick diamond slitting blade. The coupon was subsequently encapsulated in paraffin wax to act as an etch resist, and once cooled, the wax coating over the central TMC strip was removed. The matrix was removed (as in the slit etch removal method) from the TMC strip with the same etchant mixture as used in the TERM experiments. The whole coupon was submerged in the etchant and although the matrix was removed more rapidly on the outer surfaces, after prolonged exposure (approximately 16 hours), no remaining matrix was visible between the plies. The ends of the etched fibres were then viewed using a confocal microscope (Zeiss LSM 800). The microscopy established that the fibres of the inner 0° ply were on average 24 μm longer than the fibres of the outer ply over the 24 mm length i.e. the fibres of the inner ply were approximately 0.1% longer than the fibres of the outer ply; Figures 11 and 12 show overlaid images of the two sets of fibres.

Figure 11. Confocal microscopy images of the TERM exposed axial plies (LHS: side on, RHS: end on); the subsurface axial ply is shaded red, while the surface ply is shaded blue.

Figure 12. A confocal micrograph with false colour showing the difference in extension (24 μm) between the top ply (shaded blue) and subsurface axial ply (shaded red).

Consequently, the bimodal distribution of fibre lengths seen in Figure 10 is the consequence of different residual thermal strains in the axial plies, with fibres in the inner 0° plies having a residual compressive strain approximately 0.1% higher than fibres in the outer plies; consequently, when the matrix is removed, the fibres in the inner plies are longer. It is possible that the bimodal distribution of axial fibres strains within the cross-

ply laminate is principally due to the outer plies having a free surface for which relatively unhindered movement can occur in HIP processing (e.g. by creep deformation during cooling), whilst the inner plies have a greater constraint and therefore experience greater residual strains.

Taken overall, the average thermal residual strain for the 00 fibres of the cross-ply composite (εresf ≈ - 0.32%) is significantly higher than in the unidirectional composite (εresf ≈ - 0.13%). This higher residual compressive strain in the axial fibres of the cross-ply coupons is reflected in the axial failure strains for the cross-ply and unidirectional composites. The axial strain to failure of both types of TMC is dominated by the behaviour of the axial fibres; consequently, the strain-to-failure of the cross-ply coupons was approximately 1.1%, whereas the UD coupons failed at a strain of about 0.9%, as shown in Figure 1. The higher strain-to-failure of the cross-ply coupons (by 0.2%) reflects the higher residual compressive strain of the 0° fibres in the cross-ply coupons compared to the UD coupons (also about 0.2%). The total etch removal method was also applied to cross-ply coupons. Again, samples of one hundred axial fibres were selected from each of the three coupons for measurement under the shadowgraph. As the residual stress state in the cross-ply will be different from the UD specimen, a different fibre residual strain was expected. Figure 10 shows the residual strains for the three specimens and it is clear that there is a bi-modal distribution. Average results for the three coupons are shown in Table 2 where the average residual strain is about - 0.32 ± 0.08%. However, Figure 10 shows peaks in the average strain at about and there are peak average strains at strains of about - 0.23 ± 0.03% and - 0.37 ± 0.04%. The presence of the two peaks warranted further investigation.

Prediction of stress-strain curves using FE modelling

Introduction

To investigate the build-up of residual stresses, and the subsequent stress strain response of the unidirectional and cross-ply composites in tension and compression, simple three-dimensional representative volume element (RVE) models were created. The modelling had several purposes. First, if the unidirectional test data could be replicated closely, it would increase confidence in the material properties used in the FE models. Secondly, if the radial residual fibre stresses in the cross-ply model were found to be removed by tensile interfacial stresses at low applied strains, then this would help to explain the initial non-linearity in the tensile stress-strain response of the cross-ply coupons; consequently, the disparity between the simple rule-of-mixtures approach to predicting the stress-strain behaviour and the actual response of the cross-ply laminates could be understood.The consequence of the fabrication technique and processing was that the composite fibre architecture approximated well to a simple repeating array (see Figures 4 and 5) and this was represented in the FE model with a rectangular RVE model as a reasonable simplification of the material structure. Dimensions for the RVE models were obtained by measuring the average fibre spacing in micrographs of the composite. The RVE

models used to represent the composites consisted of 1/4 of a unidirectional cell and 1/16 of a cross-ply cell (Fig 13 and 14 respectively).

Figure 13. Caption: 3D RVE for the unidirectional TMC.

Figure 14. 3D RVE for the cross-ply TMC.

Constituent material properties for the models were available from prior in-house testing and literature; Table 3 shows room temperature properties, and Table 4 shows the yield strength and Young’s modulus of the matrix as a function of temperature.

Table 3. Physical properties of fibre and matrix constituents. (Strength and modulus data from [12] and CTE data from [28])

Material Yield

strength

(MPa)

UTS (MPa) Young’s modulus

(GPa)

Post-yield

modulus

(GPa)

Poisson’s

ratio

Avg. CTE

(m/m C-1)

Ti-3Al-2.5V 520 675 106 0.5 0.39 9.9 x10-6

SM3156 N/A 3800 400 N/A 0.17 4.5 x 10-6

Table 4. Ti-3Al-2.5V matrix properties at temperature ([28]).

Temperature (°C) Yield strength (MPa) Young’s modulus (GPa)

22 520 106

150 434 96

350 269 83

400 255 82

450 244 77

500 225 74

550 201 50

600 153 40

Finite-element model description

Symmetry boundary conditions were applied to three orthogonal faces of the 3D RVE models and nodal coupling equations were applied to the three opposing faces so that all nodes on a face were only allowed to move perpendicular to their parent face and by the same amount. This setup forced the faces of the RVE to remain planar, thus simulating the effect of the RVE being at the centre of a much larger region of composite. In order to simulate the effect of thermal residual stresses on the stress-strain curves, a first approximation of the lock-in temperature was derived from the TERM measurements on the unidirectionally reinforced TMCs.

As discussed above, due to differences in the CTE between the matrix and the fibre, upon liberation of the fibre from the matrix, the fibre will relax (elongate) as residual stresses are released. At the lock-in temperature, it is known that the coupon is longer than at room temperature, due to the positive coefficient of thermal expansion of the composite (c). The length of the coupon at the lock-in temperature (LcLI) can therefore be written in terms of the coupon length at room temperature (LcRT) as

LcLI=LcRT (1+αc ∆ T ) (7)Here, ΔT is the increase in temperature from ambient to the lock-in temperature and αc is the coefficient of thermal expansion of the coupon (αc = 6.6 x 10-6 K-1).As the coefficient of thermal expansion of the fibre is known (αf = 4.5 x 10-6 K-1), the length of the free fibre at room temperature (LfRT) can be expressed as

LfRT =LcLI (1−α f ∆ T ) (8)Substituting equation (7) into (8) gives

LfRT =(L¿¿cRT (1+αc ∆ T )) (1−α f ∆ T )¿ (9)Solving equation (9) for ΔT, using fibre measurements obtained from the UD TERM experiments, the lock-in temperature was found to be 667 °C above ambient. This lock-in temperature is in agreement with other authors who have used temperatures between 500-700 °C as the lock-in temperature [3, 5, 9, 17-19]. The lower end of this temperature range is thought to correspond to the temperature at which stress-relief through creep stops being significant in a titanium alloy [29]. Cool-down from HIPing typically occurs over a period of hours and would facilitate such creep conditions.

The stress strain response of the representative volume element models was obtained by applying a displacement condition to a node coupled face in-plane with a fibre end. The displacement control was deactivated until the thermal drop had been implemented in the first time-step, thus simulating the build-up of thermal residual stress. After the introduction of residual stresses to the model, the displacement control was set to increase incrementally over an arbitrary period of time (10 seconds). Lastly, the applied stress was calculated by dividing the axial reaction force by the cross-sectional area of the RVE. The applied stress was then plotted against the total strain to give the stress-strain response.

Predictions of stress-strain behaviour

Figure 15. Axial residual strain in the fibre and matrix upon cooldown.

The FE model for the UD TMC predicted a residual axial fibre strain of - 0.132% (Fig 15), which is in excellent agreement with the TERM value of - 0.132%, and in reasonable agreement with the value derived from the tensile and compressive stress/strain curves of - 0.104 % (Table 5).

Table 5. Averaged residual strains and deduced stresses.

TERM approach Stress-strain

curve derivedFinite Element Analysis Predictions

Panel Fibre Strain

(%)

Fibre Strain

(%)

Fibre Strain

(%)

Fibre Stress

(MPa)

Matrix Stress

(MPa)

[0]8 - 0.132 - 0.104 -0.132 -589 320

[0,90]2s - 0.318 N/A -0.194 -820 243

Figure 16. A comparison of tensile behaviour of the uniaxial 3D RVE and experimental test data.

Figure 17. A comparison of compressive behaviour of the uniaxial 3D RVE and experimental test data.

Figure 16 shows a comparison of the experimentally obtained tensile stress-strain curves against a curve predicted by the FE modelling for the UD TMC. Although there is reasonable agreement, the FE model predicts a departure from linearity approximately 250 MPa lower than found experimentally; this is also found for the compression stress-strain curves (Figure 17). Reductions of the same magnitude predicted for the macroscopic yield for both tension and compression loading suggest that this discrepancy is not due to the use of an incorrect ‘lock-in’ temperature, but rather an indication that the effective matrix yield strength in the composite is significantly higher than anticipated from monolithic experimental data. When component stresses upon cool-down to room temperature are investigated in the FE model, the von-Mises stress state in the UD model is seen to exceed the 520 MPa yield strength (as taken from Figure 7) of as-HIPed Ti-3-2.5 in localised areas (Fig 18). Hence, it appears likely that local constraint in the matrix between the fibres is delaying the onset of yield.

Figure 18. Von Mises stress contours resulting from cool-down in the unidirectional 3D RVE.

The FE modelling of the cross-ply (CP) composite predicts a residual axial fibre strain of - 0.194%; this is higher than the UD residual strain prediction (- 0.132%) as expected, and in reasonable agreement with the one-dimensional analytical model prediction of - 0.17%. Although the predicted residual strain is in reasonable agreement with the measured inner 00 ply fibre residual strain of the [0/90]2s TMC (about - 0.23%), it is significantly lower than the averaged residual strain for all of the 00 fibres i.e. - 0.32%. Further work on the differences in the residual strains between the inner and outer plies of the [0/90]2s TMC will be required to understand this discrepancy.

The FE analysis of the cross-ply 3D-RVE model was also used to predict the tensile and compressive stress/strain behaviour and this is compared with the rule-of mixtures approach and the experimental data for both tension and compression loading in Figures 19 and 20.

Figure 19. A graph showing the cross-ply tensile test data, the cross-ply 3D RVE response and the weighted ROM prediction.

Figure 20. A graph showing the cross-ply compressive test data, the cross-ply 3D RVE response and the weighted ROM prediction.

Despite reasonable agreement between the CP models, similar discrepancies are noted to the UD RVE models, with the knees occurring at stresses approximately 250 MPa lower than the experimental data. The weighted ROM approach is adequate for an initial approximation of the cross-ply behaviour, but fails to accurately replicate the low strain behaviour and the post-knee tangent modulus of the experimentally obtained curves. It is apparent that the role of thermal residual stresses and damage accumulation play a significant role in the cross-ply stress-strain behaviour, and this is where the limitations of the weighted rule-of-mixtures approach are realised. The thermal residual stresses induced in the matrix due to the mismatch in CTE act to apply a compressive radial stress to the fibres in the axial and transverse directions. For increasing load, debonding is possible for the transverse fibres only once the radial compressive stresses have been overcome, and this debonding may be responsible for the initial knee in the stress-strain curve. That a similar debonding does not occur for the axial fibres (since the transverse stresses do not oppose the thermal residual stresses for the axial fibres) is supported by very limited fibre pull-out of the 00 fibres on the fracture surfaces of the cross-ply TMC.

Concluding remarksIn this work, the total etch removal method (TERM) has been used to measure the thermal residual strains (and hence stresses) for the 00 fibres in unidirectional and cross-ply titanium matrix composites. These results were then compared to estimates of

residual strain obtained from mechanical test data and finite element analysis. For unidirectional composites, excellent agreement was found between the measured residual strain using TERM and finite-element predictions; the fibre strain derived from analysis of the stress-strain curves in tension and compression was 30% lower. For the cross-ply ([0/90]2s) laminates, the thermal residual strain in the 00 fibres was predicted using the FE analysis to be significantly lower than the average found for all of the 00 fibres in the laminate. However, a bi-modal distribution of residual strains was found, with the inner 00 plies experiencing a higher compressive strain in the manufactured composite than the outer 00 plies.

In addition, the results suggest that stress-strain curves for 00 plies and 900 plies can be combined using the rule-of-mixtures to give a reasonable approximation of the behaviour cross-ply ([0/90]2s) TMC, even though the approach obviously ignores: (i) the complexity of constrained yield in the matrix; (ii) the possibility of fibre/matrix debonding in the 900 ; and (iii) the differences in the thermal residual strains between the inner and outer plies of the laminate which have been revealed using the TERM approach.

AcknowledgementsThe authors of this paper would like to gratefully acknowledge the financial support of InnovateUK in part funding the NGMP programme which enabled initial testing. The authors would also like to thank Terry McCaul of Airbus Defence and Space for coordinating compression testing, and Jerry Lord of NPL for his assistance in obtaining mechanical property data for the TMCs. The lead author, Gerald Watt, would also like to thank the EPSRC for their continued support of his EngD work under the MiNMAT Industrial Doctorate Centre at the University of Surrey.

Declaration of conflicting interestsThe authors declare that there is no potential conflict of interest with respect to the research, authorship, and/or publication of this article.

Funding statementThis work was supported by the Engineering and Physical Sciences Research Council (EPSRC) [EP/GO37388/1].

Supplementary MaterialsDue to the commercially sensitive nature of the research materials supporting this publication, not all of the data can be made publicly available. Please contact the author for further details.

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